Fourier series-based automatic generation system and method for multi-variable fuzzy systems

ABSTRACT

A method of automatically generating a multi-variable fuzzy inference system using a Fourier series expansion. Sample sets are decomposed into a cluster of sample sets associated with given input variables. Fuzzy rules and membership functions are computed individually for each variable by solving a single input multiple outputs fuzzy system extracted from the set cluster. The resulting fuzzy rules and membership functions are composed and integrated back into the fuzzy system appropriate for the original sample set with a minimal computational cost. In addition, an overall system error can be related to errors at each stage of decomposition and composition, enabling error bounds or accuracy thresholds for each stage to be specified and ensuring the final precision of the resulting fuzzy system on the original sample set.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention is related to the field of fuzzy logic and,more particularly, to a system and method for automatically generating amulti-variable fuzzy inference system from given sample sets.

[0003] 2. Description of the Related Art

[0004] It is well known that the fuzzy system is extremely effective inapproximating any continuous function on a compact set. The fuzzy systemmost often provides a powerful alternative to the conventional controlsystem, particularly for complex or ill-defined systems where the systemis not easily controllable by a conventional controller. This is sobecause the fuzzy system is capable of capturing the approximate,qualitative aspects of human knowledge and reasoning needed in thecontrol system. Accordingly, fuzzy systems have recently found anincreasingly wider range of applications in industrial applications,household appliances as well as in financial analysis. The applicationscan include various manufacturing processes, robotics, consumer productssuch as heat exchangers, warm water pressure control, aircraft flightcontrol, robot control and manipulation, car speed control, powersystems and nuclear reactor control, control of a cement kiln, focusingof a camcorder, climate control for buildings, train scheduling, patternrecognition and system modeling, stock trading on a stock exchange andinformation retrieval, to mention only a few.

[0005] It is normally the domain human experts who manually code andrefine the fuzzy systems. The performance of a fuzzy system dependscritically on the depth of the experts' understanding of the domainknowledge. Accordingly, the cost of developing a fuzzy system hasgreatly increased, creating a bottleneck in the application of manycomplex fuzzy systems in new areas. This has motivated an intensivestudy of the automatic generation of fuzzy systems, and the subject hasoften been studied as a data-driven, self-adaptive problem.

[0006] In implementing self-adaptive problems by fuzzy systems, one ofthe biggest problems encountered is what is referred to as a dimensionalexplosion predicament which is associated with an exponential explosionin the required computational time as the number of variables increases.This is so because the relationship among the multi-variables of thefuzzy rules and membership functions is often quite complicated and moreoften than not is nonlinear. Because of this, almost all of the existingcommercial works available can only deal with a relatively small numberof input variables, typically a maximum of five.

[0007] In “A New Approach for the Automatic Generation of MembershipFunctions and Rules of Multi-Variable Fuzzy System”, (Proceedings ofIEEE 1995 International Conference on Neural Networks, Perth, Australia,pp. 1342-1346), Chen et al. presents a new algorithm, PolyNeuFuz, basedon polynomial approximation. PolyNeuFuz was later improved to ParNeuFuz,as discussed by Chen et al. in “A New Scheme for an Automatic Generationfor Multi-Variable Fuzzy Systems”, to be published in Fuzzy Sets andSystems, 120 (2001), no. 2, pp. 143-149. ParNeuFuz can, in principle, berun in parallel processing if the independence of the input variables isexploited.

[0008] Both PolyNeuFuz and ParNeuFuz solve the problem of generatingmulti-variable fuzzy systems by decomposing the problem to a solution ofsingle input, multiple output fuzzy systems. The ParNeuFuz algorithmexploits a polynomial expansion in approximating the given sample sets,and the time complexity of the algorithms does not depend too heavily onthe number of variables used. In principle, both of these algorithms canbe applied to generate a fuzzy system with a large number of variables.PolyNeuFuz and ParNeuFuz successfully decompose a self-adaptive problemof multi-variable systems into several steps, and into parallel stepsfor ParNeuFuz in particular, thus considerably reducing thecomputational complexity. However, unlike the Fourier series-basedapproach which is able to provide an accurate error estimate in each ofthese steps, in decomposition as well as in composition, errorestimation or prediction is not easy using the PolyNeuFuz and ParNeuFuzmethods.

[0009] Accordingly, a need exists for a scheme capable of generating amulti-variable fuzzy inference system that does not encounter thedimensional explosion predicament known in the prior art, while enablingerror estimation and prediction.

SUMMARY OF THE INVENTION

[0010] In view of the foregoing, one object of the present invention isthe development of a new Fourier series-based automatic generationscheme of a multi-variable fuzzy inference system that avoids adimensional explosion predicament.

[0011] Another object of the invention is to provide a method that iscapable of generating and computing the fuzzy rules and membershipfunctions for each variable independently of the other variables.

[0012] A further object of the invention is to provide a Fourierseries-based decomposition supporting error analysis that can specifythe error bounds or accuracy thresholds at each stage of decompositionand composition to ensure the final precision of the resulting fuzzysystem on the original sample set.

[0013] In accordance with this and other objects, the present inventionpresents a novel method of automatically generating a multi-variablefuzzy inference system for any number of practical applications. Themethod exploits the Fourier series expansion in decomposing the samplesets into simplified tasks of generating a fuzzy system with a singleinput variable independent of the other variables. The method begins bydecomposing a sample set, e.g., Λ, into an accumulation of a number ofset clusters associated with the given input variables, and computingthe fuzzy rules and membership functions for each variable,independently of the other variables, by solving a single input multipleoutputs fuzzy system extracted from the set cluster. The resultingdecomposed fuzzy rules and membership functions are integrated back intothe fuzzy system appropriate for the original sample set Λ, requiringonly a moderate computational cost.

[0014] The method described in this invention can be used to obtain astable fuzzy system by retaining any specified accuracy of the resultingmulti-variable function on the original sample set. In other words, bytaking advantage of the fact that the decomposition is based on aFourier series, a careful error analysis indicates the relationshipbetween an overall system error and errors at each stage ofdecomposition and composition so that error bounds or accuracythresholds of each of these steps may be specified to ensure the finalprecision of the resulting fuzzy system on the original sample set.

[0015] These and other features of the invention, as well as many of theintended advantages thereof, will become more readily apparent whenreference is made to the following description taken in conjunction withthe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016]FIG. 1 is a diagram of a neural network structure used forgenerating fuzzy systems of single input and multiple outputs, inaccordance with the present invention; and

[0017]FIG. 2 is a flow diagram of the method for the automaticgeneration of multivariable fuzzy system from a sample set, inaccordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0018] In describing a preferred embodiment of the invention illustratedin the drawings, specific terminology will be resorted to for the sakeof clarity. However, the invention is not intended to be limited to thespecific terms so selected, and it is to be understood that eachspecific term includes all technical equivalents which operate in asimilar manner to accomplish a similar purpose.

[0019] To begin, the main theoretical supports for the present inventionwill be summarized, followed by a discussion of two supplementingmethods. Throughout this document, the number of samples is referred toas N, and the number of input variables is referred to as m, with both Nand m being finite.

[0020] The main theoretical supports for the present invention includethe following two theorems.

[0021] Theorem 1

[0022] For any sample set Λ={({overscore (X)}_(j),Y_(j))|j=1,2, . . .,N}, where {overscore (X)}_(j)=(X_(1,j),X_(2,j), . . . ,X_(m,j)) andL_(i)≦X_(i)≦H_(i) for all i, there exists a set{a_({overscore (s)})≠0|{overscore (s)}=(s₁,s₂, . . . ,s_(2m))∈S,s₂,s₄, .. . s_(2m)∈{0,1}, with s₁, s₃, . . . ,s_(2m−1) being non-negativeintegers}, and a sample set cluster A={A_({overscore (s)})={({overscore(X)}_(j),F_({overscore (s)}j)|j=1,2, . . . ,N}|{overscore(s)}=(s₁,s₂,s₂, . . . ,s_(2m))∈S}, such that${{Y_{j} = {\sum\limits_{\overset{\_}{s} \in S}F_{\overset{\_}{s}j}}},\quad {where}}\quad$$F_{\overset{\_}{s}j} = {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}{{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}.}}}$

[0023] Theorem 2

[0024] Suppose that the single input multiple outputs sample set {tildeover (B)}_(i)={(X_(i,j),{f_({overscore (s)},i,j)|{overscore(s)}∈S})|j=1,2, . . . ,N} (|f_({overscore (s)},i,j)|≦1) can be describedby the following fuzzy system MFS_(i), with a maximum error for eachoutput and the sum of errors for each output being given bymax_(j∈{1,2, . . . ,N})|O_(fuzzy)(X_(i,j))−f_({overscore (s)},i,j)|≦ε₁,${\sum\limits_{j = 1}^{n}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq {ɛ_{2}\quad {\left( {ɛ_{1} < {{1/m}\quad {and}\quad ɛ_{1}} < ɛ_{2} < {N \cdot ɛ_{1}}} \right).}}$

[0025] Fuzzy system MFS_(i) is defined by membership functions, fuzzyrules and defuzzification method. According to the membership functionsfor each t_(i),t_(i)=1,2, . . . ,τ_(i), the membership function forvariable x_(i) at fuzzy interval S_(i,t,) is μ_(i,t) _(i) . The fuzzyrules provide that for each t_(i), t_(i)=1,2, . . . ,τ_(i), and if X_(i)is S_(i,t) _(i) then the {overscore (s)}-thoutput=O_({overscore (s)},i,t) _(i) . According to the defuzzificationmethod,${\overset{\_}{s} - {{th}\quad {Output}\quad O_{fuzzy}}} = {\sum\limits_{t_{i} = 1}^{\tau_{j}}{{\mu_{i,t_{i}}\left( X_{i} \right)} \cdot O_{\overset{\_}{s},i,t_{i}}}}$

[0026] Then the fuzzy sample set$D = \left\{ {{{\left( {{\overset{\_}{X}}_{j},{\sum\limits_{\overset{\_}{s} \in S}\left( {{a_{\overset{\_}{s}} \cdot f_{\overset{\_}{s},1,j} \cdot f_{{\overset{\_}{s},2,j}\quad}}\ldots \quad f_{\overset{\_}{s},m,j}} \right)}} \right)j} = 1},2,\ldots \quad,N} \right\}$

[0027] (|a_({overscore (s)})|≦Q) can be described by the following fuzzysystem MFS, with its maximum error and total error being given,respectively, by${{{\max\limits_{j\quad \in {\{{1,2,\quad \ldots \quad,\quad N}\}}}{{{O_{fuzzy}\left( {\overset{\_}{X}}_{j} \right)} - {\sum\limits_{\overset{\_}{s} \in \quad S}\left( {{a_{\overset{\_}{s}} \cdot f_{\overset{\_}{s},{1.j}}}f_{\overset{\_}{s},2,j}\quad \ldots \quad f_{\overset{\_}{s},m,j}\quad \ldots \quad f_{\overset{\_}{s},m,j}} \right)}}}} \leq {Q{S}\left( {{m\quad ɛ_{1}} + {O\left( {\frac{1}{2}\left( {m\quad ɛ_{1}} \right)^{2}} \right)}} \right)}},{and}}\quad$${\sum\limits_{j = 1}^{N}{{{O_{fuzzy}\left( {\overset{\_}{X}}_{j} \right)} - {\sum\limits_{\overset{\_}{s} \in S}\left( {{a_{\overset{\_}{s}} \cdot f_{\overset{\_}{s},1,j}}f_{\overset{\_}{s},2,j}\quad \ldots \quad f_{\overset{\_}{s},m,j}} \right)}}}} \leq {\left( {1 + {\frac{1}{2}m\quad ɛ_{1}} + {O\left( {\frac{1}{6}\left( {m\quad ɛ_{1}} \right)^{2}} \right)}} \right){mQ}{S}{ɛ_{2}.}}$

[0028] The fuzzy system MFS is also defined by membership functions,fuzzy rules and defuzzification method. According to the membershipfunctions, for each pair of (i,t_(i)),i_(i)1,2, . . . ,m;t_(i)=1,2, . .. ,τ_(i), the membership function for variable x_(i) at fuzzy intervalS_(i,t) _(i) is μ_(i,t) _(i) . The fuzzy rules provide that for eachm-tuple (t₁,t₂, . . . ,t_(m)), t₁=1,2, . . . ;τ_(i);t₂=1,2, . . . ,τ₂; .. . ;t_(m)=1,2, . . . ,τ_(m), if X₁ is S_(1,t) _(i) and X₂ is S_(2,t) ₂and . . . and X_(m) is S_(m,t) _(m) then${Output} = {O_{t_{1},t_{2},\quad \ldots \quad,t_{m}} = {\sum\limits_{\overset{\_}{s} \in \quad S}{\left( {a_{\overset{\_}{s}} \cdot {\prod\limits_{i = 1}^{m}O_{\overset{\_}{s},i,t_{i}}}} \right).}}}$

[0029] According to the defuzzification method,$O_{fuzzy} = {\sum\limits_{\underset{{i = 1},2,\quad \ldots \quad,m}{{t_{i} = 1},2,\quad \ldots \quad,\tau_{i}}}\left( {\prod\limits_{l = 1}^{m}{{\mu_{l,t_{i}}\left( X_{l} \right)} \cdot O_{t_{1},t_{2},\quad \ldots \quad,t_{m}}}} \right)}$

[0030] Theorem 1 ensures the decomposition of the sample set into anumber of sample set clusters, which form the basis for thedecomposition process of the method of the present invention. That is,Theorem 1 ensures the existence of set cluster$A = \left\{ {{A_{\overset{\_}{s}} = \left\{ {{\left( {{\overset{\_}{X}}_{j},{F_{\overset{\_}{s}j}\left. {{j = 1},2,\quad \ldots \quad,N} \right\}}} \right.\overset{\_}{s}} = {\left( {s_{1},s_{2},\quad \ldots \quad,s_{2m}} \right) \in S}} \right\}},{{{such}\quad {that}\quad Y_{j}} = {\sum\limits_{\overset{\_}{s} \in \quad S}{F_{\overset{\_}{s}j}.}}}} \right.$

[0031] Theorem 2 ensures the construction and composition of decomposedsingle input fuzzy systems to recover the fuzzy system on the originalsample set Λ. That is, Theorem 2 shows that, once the membershipfunctions and fuzzy rules of each variable X_(i) are computed on sampleset${\overset{\sim}{B}}_{i} = \left\{ {{{\left( {X_{i,j},\left\{ {{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}\left. {\overset{\_}{s} = {\left( {s_{1},s_{2},\quad \ldots \quad,s_{2m}} \right) \in S}} \right\}} \right)} \right.j} = 1},2,\ldots \quad,N} \right\}$

[0032] the fuzzy system appropriate for A may be constructed byfollowing the steps of Theorem 2.

[0033] The present invention also makes use of two supplementingmethods, which are the decomposition method and the method for obtainingthe fuzzy systems on sample sets {tilde over (B)}_(i) (i=1,2, . . . ,m).They will be described below.

[0034] Decomposition Method

[0035] Knowledge on the sample set as well as on the smoothness of anobject surface plays an important role in the analysis used by thepresent invention. Suppose that the object system does not includehigher components than 2πτ_(i)/(U_(i)−L_(i)) for each variable X_(i).Then it is possible to select an S={(s₁,s₂, . . . ,s_(2m))} wheres₁≦τ₁,s₃≦τ₂, . . . ,s_(2m−1)≦τ_(m). An optimal approximation to thesample set can be obtained by minimizing the following function:$\begin{matrix}{E = {\frac{1}{2}{\sum\limits_{j = 1}^{N}\left( {Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\quad \ldots \quad,\quad s_{2m}})}\quad \in \quad S}}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}} \right)^{2}}}} & (1)\end{matrix}$

[0036] This is equivalent to finding the roots of the equations∂E/∂a_({overscore (s)})=0. Some care must be taken in the process sothat all terms of equal wave numbers are collected into one term, notretaining any two identical terms differing only in coefficients. Forexample, the two terms${a_{({0,0,1,0})}{{\cos (0)} \cdot {\cos \left( {\frac{2\pi \quad s_{1}}{U_{1} - L_{1}} \cdot \left( {X_{2} - \frac{U_{1} - L_{1}}{2}} \right)} \right)}}\quad {and}}\quad$$a_{({0,1,1,0})}{{\cos \left( \frac{\pi}{2} \right)} \cdot {\cos \left( {\frac{2\pi \quad s_{1}}{U_{1} - L_{1}} \cdot \left( {X_{2} - \frac{U_{1} - L_{1}}{2}} \right)} \right)}}$

[0037] should be combined into$a_{({0,0,1,0})}^{\prime}{{\cos (0)} \cdot {\cos \left( {\frac{2\pi \quad s_{1}}{U_{1} - L_{1}} \cdot \left( {X_{2} - \frac{U_{1} - L_{1}}{2}} \right)} \right)}}$

[0038] before the coefficients of the terms of Fourier series aredetermined. After the identical terms have been combined, suppose atthis time, S=({overscore (s)}₁,{overscore (s)}₂, . . . ,{overscore(s)}_(τ)), with {overscore (s)}_(k)=(s_(k) ₁ ,s_(k) ₂ ,s_(k) _(2m) ),k=1,2, . . . ,τ, the a_({overscore (s)}) _(k) can be obtained by anyStandard Gaussian Elimination or by the following computation:$\begin{matrix}{{a_{{\overset{\_}{s}}_{k}} = {\frac{{T_{p,q}^{k}}_{\tau \times \tau}}{{J_{p,q}}_{\tau \times \tau}}\quad {where}}}{J_{p,q} = {\sum\limits_{j = 1}^{N}{\prod\limits_{i = 1}^{m}{\left( {{\cos \left( {{\frac{2\pi \quad s_{p_{{2i} - 1}}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{p_{2i}} \cdot \frac{\pi}{2}}} \right)}{\cos \left( {{\frac{2\pi \quad s_{q_{{2i} - 1}}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{q2i} \cdot \frac{\pi}{2}}} \right)}} \right)\quad {and}}}}}{T_{p,q}^{k} = \left\{ \begin{matrix}{\sum\limits_{j = 1}^{N}{Y_{j}{\prod\limits_{i = 1}^{m}{\cos \left( {{\frac{2\pi \quad s_{k_{{2i} - 1}}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{k_{2i}} \cdot \frac{\pi}{2}}} \right)}}}} & {{{if}\quad q} = k} \\{\sum\limits_{j = 1}^{N}{\sum\limits_{i = 1}^{m}\left( {\cos \left( {{\frac{2\pi \quad s_{p_{{2i} - 1}}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{p_{2i}} \cdot \frac{\pi}{2}}} \right)} \right.}} & {{{if}\quad q} \neq k} \\{\left. {\cos \quad \left( {{\frac{2\pi \quad s_{q_{{2i} - 1}}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{q2i} \cdot \frac{\pi}{2}}} \right)} \right)\quad} & \quad\end{matrix} \right.}} & (2)\end{matrix}$

[0039] In case the sample data fall exactly on the multidimensionalgrids, a standard Fast Fourier Transform method can be employed to speedup this process.

[0040] Generating Single Input Multiple Outputs Fuzzy Systems

[0041] With reference to the neural network of FIG. 1, when each of thelink weights b_(it,k) between the second layer neuron and the thirdlayer neuron can be interpreted as a rule: “if X_(i) is S_(i,t) then theoutput of f_(k) is b_(it,k)”, then the relationship between the inputand outputs of the neural network,f_(k)=b_(i1,k)μ_(i1)(X_(i))+b_(i2,k)μ_(i2)(X_(i))+ . . . +b_(iτ) _(i),_(k)μ_(iτ) _(i) (X_(i)) can be described by the fuzzy system S, inwhich the membership functions provide that for each t, where t=1,2, . .. ,τ_(i), the membership function of fuzzy interval S_(i,t) is μ_(i,t);and the fuzzy rules provide that for each pair of (t, k) where t=1,2, .. . ,τ_(i); k=1,2, . . . ,τ, if X_(i) is S_(i,t) then the output off_(k)=b_(it,k). According to the defuzzification method, output of$f_{k} = {\sum\limits_{t = 1}^{\tau_{i}}{{\mu_{i,t}\left( X_{i} \right)}{b_{{it},k}.}}}$

[0042] Thus, the task of obtaining the membership functions and fuzzyrules of each variable X_(i) reduces to training the neural network ofFIG. 1 using the set {tilde over (B)}_(i) as its sample set. Themembership functions of each variable can either be of “straight” typeor continuous type. The Steepest Descent (SD) method such as that taughtby Cauchy, “Methode Generale pour la Resolution des Systems d'EquationsSimultaneos”, Comp. Rend. Acad. Sci. Paris, pp. 536-538, 1847, can beused to train these individual networks.

[0043]FIG. 2 summarizes the method of the present invention for theautomatic generation of a multivariable fuzzy system from a sample set.To demonstrate the inventive algorithm, which will be referred to hereinas algorithm FouNeuFuz, we will begin by supposing that we need todevelop the fuzzy system on sample set A with the maximum error and thetotal error of ε₁ and ε₂ respectively, such that max${{\max_{j \in {\{{1,2,\ldots,N}\}}}{{{O_{fuzzy}\left( {\overset{\_}{X}}_{j} \right)} - Y_{j}}}} \leq ɛ_{1}},{{{and}\quad {\sum\limits_{j = 1}^{n}{{{O_{fuzzy}\left( {\overset{\_}{X}}_{j} \right)} - Y_{j}}}}} \leq {ɛ_{2}.}}$

[0044] As shown in FIG. 2, the method begins by decomposing the sampleset Λ, step 100, into the sample set clusterA={A_({overscore (s)})|{overscore (s)}εS} by minimizing Equation (1)using Gaussian Elimination method or by Equation (2). At step 110, themaximum error and the total errors are calculated:${ɛ_{f1} = {\max_{j \in {\{{1,2,\ldots,N}\}}}{{Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\ldots,s_{2m}})} \in S}}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}}}}},{ɛ_{f2} = {\sum\limits_{j = 1}^{N}{{{Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\ldots,s_{2m}})} \in S}}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}{\cos \quad \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \quad \quad \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}}}.}}}$

[0045] The error requirement is then verified, step 120, by making surethat ε_(f1)≦ε₁ and ε_(f2)≦ε₂. If it is not so, terms with higherfrequency are added and the calculation of steps 100 and 110 arerepeated until satisfied.

[0046] At step 130, for each variable X_(i), the fuzzy rules andmembership functions on sample set {tilde over (B)}_(i) are obtained bytraining the neural network as shown in FIG. 1 by the steepest descent(SD) method, where we choose${{\overset{\sim}{B}}_{i} = \left\{ {{{\left( {X_{i,j},\left\{ {{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}\left. {\overset{\_}{s} = {\left( {s_{1},s_{2},\ldots \quad,s_{2m}} \right) \in S}} \right\}} \right)} \right.j} = 1},2,\ldots \quad,N} \right\}},$

[0047] the terminating condition of training each network is set as: max${{\max_{j \in {\{{1,2,\ldots,N}\}}}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq ɛ_{s1}} = {\min \left\{ {\frac{1}{m},\frac{ɛ_{1} - ɛ_{f1}}{{mQ}{S}}} \right\} \quad {and}}$${{\sum\limits_{j = 1}^{n}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq ɛ_{s2}} = \frac{ɛ_{2} - ɛ_{f2}}{{mQ}{S}\left( {1 + {\frac{1}{2}m\quad ɛ_{s1}}} \right)}$

[0048] for each output of the networks, where

[0049] Q=max {|a_({overscore (s)})|}.

[0050] The fuzzy rules and the membership functions are thenaccumulated, step 140, into an integrated fuzzy system on sample set Λaccording to Theorem 2.

[0051] The invention as described can be used to obtain a fuzzy systemautomatically using given sample data, rather than to design the fuzzysystem manually as is often done in practice in the prior art. Such afuzzy system has many practical applications. For example, in theindustrial motor control field, it is necessary to model therelationship between magnetizing current and the slip frequency.However, the magnetizing current is a non-linear function of the slipfrequency, the rotor time constant, the rotor leakage factor, and anon-constant offset current. What we have at hand are only a set ofsample data, each of which indicates just the real magnetizing currentin a special situation with known slip frequency, rotor time constant,rotor leakage factor, and offset current. According to the prior art, anattempt would be made to obtain the fuzzy system manually describing therelationship, but this is extremely time consuming and difficult fornon-specialists. With the use of the novel methods described in thepresent invention, the fuzzy system can be obtained automatically anddirectly from the sample set.

[0052] Another example is for the fuzzy controller of a washing machine.Here it is necessary to control the water requirement in subsequentwashing steps based on the data of laundry load, water absorption speed,water absorption volume, and water temperature. We have only at hand thesample sets consisting of the data indicating the experts' suggestedwater requirement in sub sequential washing steps for some special(individual) situation with known laundry load, water absorption speed,water absorption volume and water temperature. With the presentinvention, it is possible to obtain the fuzzy system which can suggestthe requirement of water in sub sequential washing step (output) basedon the laundry load, water absorption speed, water absorption volume andwater temperature (Input). In the above applications, the method of thepresent invention may be used to obtain 4 inputs-1 output fuzzy systemsfrom sample data sets.

[0053] The foregoing descriptions and drawings should be considered asillustrative only of the principles of the invention. The invention maybe configured in a number of ways and is not limited by the specificconfiguration of the preferred embodiment. Numerous applications of thepresent invention will readily occur to those skilled in the art,various manufacturing processes, robotics, consumer products such asheat exchangers, warm water pressure control, aircraft flight control,robot control and manipulation, car speed control, power systems andnuclear reactor control, control of a cement kiln, focusing of acamcorder, climate control for buildings, train scheduling, patternrecognition and system modeling, stock trading on a stock exchange andinformation retrieval, to mention only a few. Therefore, it is notdesired to limit the invention to the specific examples disclosed or theexact operation shown and described. Rather, all suitable modificationsand equivalents may be resorted to, falling within the scope of theinvention.

What is claimed is:
 1. A method for the automatic generation of amultivariable fuzzy system from a sample set for use in a control systemcomprising the steps of: decomposing a sample set Λ into a sample setcluster A={A_({overscore (s)})|{overscore (s)}∈S}; calculating maximumerror and total error; verifying that the maximum error and the totalerror are equal to or greater than respective threshold values;obtaining a fuzzy system for each variable; accumulating a final fuzzysystem on the sample set Λ; and applying the final fuzzy system to thecontrol system.
 2. The method as set forth in claim 1, the step ofcalculating maximum error and total error being performed usingrespective equations,${ɛ_{f1} = {\max_{j \in {\{{1,2,\ldots,N}\}}}{{Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\ldots,s_{2m}})} \in S}}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}}}}},{ɛ_{f2} = {\sum\limits_{j = 1}^{N}{{{Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\ldots,s_{2m}})} \in S}}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}{\cos \quad \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \quad \quad \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}}}.}}}$


3. The method as set forth in claim 2, the step of verifying includingensuring that ε_(f1)≦ε₁ and ε_(f2)≦ε₂ and, in response to a negativeoutcome, including the step of adding terms with higher frequency andrepeating the steps of decomposing and calculating until ε_(f1)≦ε₁ andε_(f2)≦ε₂.
 4. The method as set forth in claim 1, the step of obtaininga fuzzy system for each variable including the steps of, for eachvariable X_(i), obtaining fuzzy rules and membership functions on sampleset {tilde over (B)}_(i) by training a neural network where${{\overset{\sim}{B}}_{i} = \left\{ {{{\left( {X_{i,j},\left\{ {{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}\left. {\overset{\_}{s} = {\left( {s_{1},s_{2},\ldots \quad,s_{2m}} \right) \in S}} \right\}} \right)} \right.j} = 1},2,\ldots \quad,N} \right\}},$

the terminating condition of training each network is set as max${{{\max_{j \in {\{{1,2,\ldots \quad,N}\}}}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq ɛ_{s1}} =},{\min \left\{ {\frac{1}{m},\frac{ɛ_{1} - ɛ_{f1}}{{mQ}{S}}} \right\}}$and${{\underset{j = 1}{\overset{n}{\quad\sum}}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq ɛ_{s2}} = \frac{ɛ_{2} - ɛ_{f2}}{{mQ}{S}\left( {1 + {\frac{1}{2}m\quad ɛ_{s1}}} \right)}$

for each output of the network, where Q=max {|a_({overscore (s)})|}. 5.A method for the automatic generation of a multivariable fuzzy systemfrom a sample set for use in a control system comprising the steps of:decomposing a sample set Λ into a sample set clusterA={A_({overscore (s)})|{overscore (s)}∈S}; calculating maximum error andtotal error using respective equations${ɛ_{f1} = {{\max \quad}_{j \in {\{{1,2,\ldots \quad,N}\}}}{{Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\ldots \quad,s_{2m}})} \in S}}^{\quad}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}\quad {\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}}}}},{ɛ_{f2} = {\sum\limits_{j = 1}^{n}{{Y_{j} - {\sum\limits_{\overset{\_}{s} = {{({s_{1},s_{2},\ldots \quad,s_{2m}})} \in S}}^{\quad}\left( {a_{\overset{\_}{s}}{\prod\limits_{i = 1}^{m}\quad {\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}}} \right)}}}}},$

verifying that ε_(f1)≦ε₁ and ε_(f2)≦ε₂ and, in response to a negativeoutcome, adding terms with higher frequency and repeating the steps ofdecomposing and calculating until ε_(f1)≦ε₁ and ε_(f2)≦ε₂; obtaining afuzzy system by each variable X_(i), including fuzzy rules andmembership functions on sample set {tilde over (B)}_(i) by training aneural network where${{\overset{\sim}{B}}_{i} = \left\{ {{{\left( {X_{i,j}\left\{ {{\cos \left( {{\frac{2\pi \quad s_{{2i} - 1}}{U_{i} - L_{i}} \cdot \left( {X_{i,j} - \frac{U_{i} - L_{i}}{2}} \right)} + {s_{2i} \cdot \frac{\pi}{2}}} \right)}\left. {\overset{\_}{s} = {\left( {s_{1},s_{2},\ldots \quad,s_{2m}} \right) \in S}} \right\}} \right)} \right.j} = 1},2,\ldots \quad,N} \right\}},$

2, . . . ,N}, the terminating condition of training each network is setas${{{\max_{j \in {\{{1,2,\ldots \quad,N}\}}}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq ɛ_{s1}} =},{\min \left\{ {\frac{1}{m},\frac{ɛ_{1} - ɛ_{f1}}{{mQ}{S}}} \right\}}$and${{\underset{j = 1}{\overset{n}{\quad\sum}}{{{O_{fuzzy}\left( X_{i,j} \right)} - f_{\overset{\_}{s},i,j}}}} \leq ɛ_{s2}} = \frac{ɛ_{2} - ɛ_{f2}}{{mQ}{S}\left( {1 + {\frac{1}{2}m\quad ɛ_{s1}}} \right)}$

for each output of the network, where Q=max {|a_({overscore (s)})|};accumulating a final fuzzy system on the sample set Λ; applying thefinal fuzzy system to the control system.